Probably still quite a bit of information is visible about you, if someone's interested in looking.
Are IP Addresses given to a phone static or not?
Not static. The IP address of your phone is dynamic (it changes regularly). The details of exactly how regularly and what the IP range will be depends on your provider. In many cases, your IP will simply appear to the internet to be located wherever your mobile provider's hardware is located, usually a nearby major or capital city. It would, though, be deducible that you're using a mobile connection and what company your internet connection is coming through. (Note that this may not actually be your phone company — low-cost carriers often piggyback on a bigger company's services at a lower priority and speed to their own customers.)
Also, how can phone companies keep track of person if they wanted to find them? For example, I don't have an account and use a pre-paid card.
If the company was interested (and their interest probably only extends to marketing purposes, but that's a powerful motivator for data collection), they can track a fair amount of information, especially if your handset has the company's software built-in (cheap prepaid handsets are usually full of branded software). If the phone company has any software on your device, then the answer may be that they can see literally everything about you and what you're doing — see, e.g., recent coverage of this problem in the tech press.
But let's assume you don't have software from the phone company on your handset. If they are interested in collecting data about you (and, if only for marketing purposes, they are), the can see and collect the following data as a part of normal business:
- The cellular tower you are connected to, and the strength of you connection (which can imply distance, and if you're indoors or outdoors)
- The details of the SIM card you've been issued, which is what actually connects you to your phone number and account
- The IMEI number of your handset or mobile device. This is issued by the manufacturer of your device, and can reveal not just the brand but also the model. In fact, for some models it could show them (from most to least likely) how much storage space you sprung for, the color of the handset, the country that it was manufactured for, or even the retailer that sold it.
It's probably also obvious, but worth pointing out:
- They can see the numbers you call, which tower you're connected to when you make the call, and how long you talk for.
- They can see who you send SMS messages to, which tower you're connected to when you send them, and the complete contents of those texts.
- They can see the origin and length of calls you receive, and the origin and contents of texts you receive. If the caller/sender is on the same network, then they have high visibility on both sides of the connection, including the cell tower the caller is connected to at the time.
The above information can reveal information about your employer(s), family, romantic interests, and daily habits, including your approximate locations at different times of day.
So far we haven't gotten into IP and data services at all, and there is already ample data to deanonymize the average person, regardless of whether you gave them account information. Also consider, your account is prepaid, but how do you prepay it? Card transaction details will be recorded.
I feel like I should note that even if the phone company isn't interested in using your personal data for marketing reasons, they still must record all the information above for legal and billing reasons.
I'm wondering how Celluar-Data works compared to the internet
Well, your cellular data provider is an ISP. So everything you know about your information security or privacy with respect to ISPs also applies. To the wider internet, your connection is no different to that of any other internet-connected device, except for as noted above that your IP can identify you as a phone company's customer. It's also worth pointing out that the wider internet cannot see any details about your cellular connection — your phone number, IMEI number, cell tower, etc.